91Ó°ÊÓ

The slope of a tangent line to a graph at any given point is basically the same as the value of the derivative at that point. It tells us how steep the line is at that point on the curve. To find it, first differentiate the function. This process provides us with a new function that represents the slope of the original function at any point on its curve.

In this exercise, we were given the function \(g(x) = \frac{x}{x-2}\). By differentiating it, we find the slope for any \(x\). Then, by evaluating this derivative at \(x = 3\), we know the slope at point \((3,3)\) is \(-2\).

This \'-2\' signifies a downward slope, meaning the tangent line decreases as it moves right across the curve at the given point.

The quotient rule is a handy tool in calculus, especially when you are dealing with functions that are divided by each other, like \(\frac{u}{v}\). The rule itself is expressed as:\[\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}\]

With the given function \(g(x) = \frac{x}{x-2}\), we identify \(u = x\) and \(v = x - 2\). Following the quotient rule, you differentiate \(u\) (which is \(1\)) and \(v\) (which is also \(1\)) and plug them into the formula. This helps us determine that the derivative \(g'(x) = \frac{-2}{(x-2)^2}\).

Remember, the quotient rule is vital for tackling fractions in derivatives and gives us the right slope expression by focusing on how each part of a fraction changes.

Evaluating the derivative at a specific point helps in determining the slope of the tangent line at that exact spot on the curve. It transforms the general expression we obtained after differentiation into a specific numeric value.

In our example, after deriving \(g'(x) = \frac{-2}{(x-2)^2}\), we plugged in \(x = 3\) and computed \(g'(3) = \frac{-2}{1} = -2\). This gives us the slope at point \((3,3)\).

This process is essential since it doesn't just give the local behavior of the function directly around the point, but effectively defines the slope of the curve precisely there, facilitating the tanget line application.

Once we have the slope of the tangent line, constructing its equation becomes straightforward. The point-slope form is particularly useful here; it is given by:\[y - y_1 = m(x - x_1)\]

Here, \(m\) is the slope we found, which is \(-2\), and \((x_1, y_1)\) is the point \((3,3)\). Substituting these into the formula, we get \(y - 3 = -2(x - 3)\).

By simplifying this expression, \(y - 3 = -2x + 6\), and then rearranging to get \(y = -2x + 9\), we find the explicit equation of our tangent line.

This line now describes exactly how the tangent behaves as it touches the curve at the given point, providing both its slope and position on a graph.